Topological dual and extended relations between networks of clathrate hydrates and Frank-Kasper phases

Clathrate hydrates are a class of ordered structures that are stabilized via the delicate balance of hydrophobic interactions between water and guest molecules, of which the space-filling network of hydrogen-bonded (H-bonded) water molecules are closely related to tetrahedrally close-packed structures, known as Frank-Kasper (FK) phases. Here we report an alternative way to understand the intricate structures of clathrate hydrates, which unveils the diverse crystalline H-bonded networks that can be generated via assembly of one common building block. In addition to the intrinsic relations and pathways linking these crystals, we further illustrate the rich structural possibilities of clathrate hydrates. Given that the topological dual relations between networks of clathrate hydrates and tetrahedral close-packed structures, the descriptors presented for clathrate hydrates can be directly extended to other ordered materials for a more thorough understanding of their nucleation, phases transition, and co-existence mechanisms.


Supplementary Note 1. Discussion on topological dual and steric connectivity
Taking A15 of Frank-Kasper (FK) phases and Type I clathrate hydrates for example, as shown in Supplementary Fig. 1a, applying Voronoi tessellation to tetrahedral close-packed (TCP) nodes will generate a different form of network. This network is composed by hydrogen-bonded (H-bonded) water molecules forming clathrate hydrates ( Supplementary   Fig. 1a); a 5 12 cage on the corner of the Type I clathrate hydrate unit cell shows the detailed arrangement of the water molecules. TCP nodes and H-bond network are two sides of the same coin. Considering the connectivity of H-bonds is equivalent to considering the arrangement of the TCP nodes.
The network of TCP nodes are generally described by Coordination Number (CN) polyhedrons and their geometry is shown in Fig. 1. The surrounding nodes of one CN polyhedron are the center or surrounding nodes for another CN polyhedron, that means two CN polyhedrons can be connected by sharing a node, an edge, a triangular face, or a region. As show in Supplementary Fig. 1b, the intersection of CN12 and CN14 in A15 phase is a region (bounded by magenta nodes and two red central nodes). The complex connectivity among CN polyhedrons makes it intractable in understanding detailed 3D structures. While the connectivity among their topological duals, Voronoi cells, is much simpler and much more defined. Mindful that these Voronoi cells actually are H-bond cages, their connectivity must satisfy the connectivity of hydrogen-bonded water molecules. The maximum number of hydrogen bonds that a water molecule can form is four, donating two via its two hydrogen atoms and accepting two via its oxygen atom. Thus, when two H-bond cages are connected, they must be connected by sharing a full face, that is the only way. If two H-bond polyhedrons connect by sharing one node (a water molecule) or one edge (a H-bond), it means the total edges converging at the shared nodes exceeds four, which is impossible for H-bonded water molecules. One can use a 2D image as an analogy, as show in Supplementary Fig. 1c,d.
To obtain a points array like Supplementary Fig. 1c, one can try to directly consider the arrangement pattern of these points. On the other hand, one can consider a framework composed of a series of triangles, as shown in Supplementary Fig. 1d. In this case, one only needs to place a node at the center of each triangle, then the same points array in Supplementary Fig. 1c automatically appears. Considering TCP nodes directly generate the points array in Supplementary Fig. 1c, and considering H-bond cages with the method shown in Supplementary Fig. 1d.
Owing to the complex connectivity among CN polyhedrons, some methods were developed for a better interpretation and exploration of the TCP network. The major network/skeleton associated with layers of nets is the best known method. As shown in Fig. 1 in the manuscript, there are two kinds of nodes in the CN polyhedrons, 5-fold and 6-fold.
The lines connecting the 6-fold nodes are called the major network/skeleton of FK phases.
Supplementary Fig. 2a shows the major network (green lines) of A15 phase. In this method, the TCP nodes are sliced into layers of nets formed by the major network. Supplementary   Fig. 2b shows the nets that establishes A15 phase, however, it is not straightforward to consider and understand the correlations across layers, while the basic building block (BBB) provide this information, as shown in Supplementary Fig. 2a, where the PC1 building block shows the connectivity among different layers of net. We see major lines penetrate the hexagonal rings of the BBB, so it also provides a way to geometrically combine major network and CN12 nodes. Moreover, for the major network/skeleton method, different FK phases have their own set of nets, which also makes it difficult to understand the structural relations among diverse FK phases. The BBB perspective has advantage in this aspect, since structures are considered through one common building block, as discussed in the manuscript.
One can find more information about major network of other type FK phases, 1,2 and other interpretation methods 3-6 of FK phases in other studies.
Overall, a simpler and much more defined connectivity among H-bond cages makes it helpful for a better understanding of the detailed 3D structures. Based on this point, the building blocks proposed in this work (through deconstruction of the H-bond network of clathrate hydrates) provide a simplified and unified way to understand intricate 3D networks of FK phases and clathrate hydrates, just like auxiliary lines used in solving complex geometric problems, which reveal the intrinsic relations and pathways linking diverse crystals.

Supplementary Note 2. Distortion of Basic building block
The BBB is not exactly the same in different clathrate hydrates in terms of geometric parameters. In Supplementary Fig. 3a, we align the BBB in Type I (red) and Type HS-I (blue) to illustrate the slightly differences and distortions.    Supplementary Fig. 10 Alternative unit cell of Type II. a,b Conventional cubic unit cell of Type II structure (F d3m). 5 12 and 5 12 6 4 are displayed in blue and orange, respectively. Cages are hidden in (b) for highlighting the pyramid building blocks. c Inverted connected pyramid building blocks. Hexagonal rings are displayed in different color (red and grey) to show the relationship between two layers, that is, the red layer six-fold axes coincide with grey layer three-fold axes, and vice-versa. d A series of p6 layers (represented by R-A building block) generated by the inverted connected pyramid building blocks. Two versions of the unit cell can be obtained from the special offset stacking of p6 layers. Magenta and green lines indicate the rhombohedral and prismatic cell boundary, respectively. e,f Geometric parameters in the offset layers. Based on the pyramid building block, one can easily infer the offset angle in the layered Type II is 54.7 • (angle between an edge and a face in the regular tetrahedron) Supplementary Table 5 Parameters of conventional unit cell and alternative unit cell of Type II structure. The cages in the "Cell content" row have consistent color code used in the Supplementary Fig. 5h.  Fig. 6h can transit to P 4 2 /mmc unit cell of Type I, which follows the mechanism described in Types I and HS-I pathway section. The red dash lines highlight the same structure. c,d An example captured in MD simulation, although the P 4 2 /mmc unit cell is incomplete.